Page 524 - Fiber Optic Communications Fund

Digital Signal Processing 505

From Eq. (11.35), we find
{ }
kN
−1 ∑ M ̂
Arg (̃y ) ≡ Δ ≅Δ . (11.36)
l
k
l
M
l=(k−1)N+1
̂
̂
Here, Δ is the phase estimate of the kth block. The signal sample ̃y is multiplied by exp (iΔ ) to obtain
k
l
k
the estimate of the transmitted signal,
̂
̂ x = ̃y exp (iΔ ). (11.37)
l l k
The computation in each block can be carried out using separate signal processors. Finally, the signal samples
in each block are combined using a multiplexer to obtain the serial data. The block size should be chosen
′
carefully. If N is too small, the impact of the noise term n in Eq. (11.32) can not be ignored. If N is too
l
large, the laser phase may drift and Δ may not remain constant within each block. The block size should
l
be optimized based on the laser linewidth.
11.5.1 Phase Unwrapping
The function Arg() in Eq. (11.36) can not distinguish between phases that differ by 2 and it returns the results
in the interval [−, ]. If the phase is + , > 0, the function Arg() returns a phase of − + . This is known
as phase wrapping and it could lead to symbol errors. Special techniques have to be used to unwrap phases.
Consider the following example: suppose Δ in the current block k is ≪ , > 0, and let Δ be roughly
l l
̂
constant over the block. From Eq. (11.36), it follows that Δ = .Now,let Δ of the next block, k + 1jump
k l
by ∕M, i.e., Δ of the (k + 1)th block is + ∕M. From Eq. (11.36) for the (k + 1)th block, we find
l
{ }
(k+1)N
−1 ∑ [ ( )] −1
Arg A exp −i M + M + = Arg{exp [−i(M + )]}
l
l
M M
l=kN+1
M −
=
M
=Δ ̂ . (11.38)
k+1
Clearly, the estimated phase − ∕M is different from the actual phase, + ∕M. This is because of
the phase wrapping done by the function of Arg(). Phase wrapping in the context of coherent optical
̂
communication has been studied in Refs. [14, 15]. Let the carrier phase prior to the unwrapping be Δ .If
k
we add 2∕M to Δ ̂ , the phase for the (k + 1)th block after the phase unwrapping is
k+1
Δ k+1 =Δ ̂ k+1 + 2 = + ∕M, (11.39)
M
which is actually the phase of the (k + 1)th block. In general, an integral multiple of 2∕M is added to the
carrier phase. The carrier phase after the phase unwrapping can be written as
̂
Δ =Δ + m2∕M, k = 1, 2,...,K, (11.40)
k
k
where ( )
Δ k−1 −Δ ̂ k
m = Floor 0.5 + . (11.41)
2∕M
Here, Floor() returns the nearest integer toward −∞. Suppose the phases of the kth and (k − 1)th block are both
. In this case, m = 0 and the phase unwrapping block of the phase estimator Eq. (11.40) does not add 2∕M.